Does wind affect the distance that sound travels

Special theory of relativity

We denote the speed of the airplane relative to the air with \ (c \), the speed of the air relative to the ground (wind) with \ (v \) and the speed of the airplane relative to the ground with \ ({c _ {{\ rm { eff}}}} \).

a) When there is no wind, \ ({c _ {{\ rm {eff}}}} = c \). Thus \ ({c _ {{\ rm {eff}}}} = 500 \ frac {{{\ rm {km}}}} {{\ rm {h}}} \) and finally \ [t = 2 \ cdot \ frac {s} {{{c _ {{\ rm {eff}}}}}} \ Rightarrow t = 2 \ cdot \ frac {{650 {\ rm {km}}}}} {{500 \ frac {{ {\ rm {km}}}} {{\ rm {h}}}}} = 2.6 {\ rm {h}} = 2 {\ rm {h}} 36 {\ rm {min}} \]

b) In cross winds, the vector diagram (Pythagoras) gives \ (c _ {\ text {eff}} = \ sqrt {c ^ 2 - v ^ 2} \). Thus \ ({c _ {{\ rm {eff}}}} = 490 \ frac {{{\ rm {km}}}} {{\ rm {h}}} \) and finally \ [t = 2 \ cdot \ frac {s} {{{c _ {{\ rm {eff}}}}}} \ Rightarrow t = 2 \ cdot \ frac {{650 {\ rm {km}}}}} {{490 \ frac {{ {\ rm {km}}}} {{\ rm {h}}}}} = 2.66 {\ rm {h}} = 2 {\ rm {h}} 40 {\ rm {min}} \]

c) The vector diagram results in \ ({c _ {{\ rm {eff}}}} = c - v \) for headwind and \ ({c _ {{\ rm {\ rm {eff}}}} = c + v for tailwind \). This gives \ ({c _ {{\ rm {eff, hin}}}} = 600 \ frac {{{\ rm {km}}}}} {{\ rm {h}}} \) and thus \ [{ t _ {{\ rm {towards}}}} = \ frac {s} {{{c _ {{\ rm {eff}} {\ rm {, towards}}}}}}} \ Rightarrow {t _ {{\ rm { hin}}}} = \ frac {{650 {\ rm {km}}}} {{600 \ frac {{{\ rm {km}}}} {{\ rm {h}}}}} = 1, 08 {\ rm {h}} = 1 {\ rm {h}} 05 {\ rm {min}} \] and \ ({c _ {{\ rm {eff, back}}}} = 400 \ frac {{ {\ rm {km}}}} {{\ rm {h}}} \) and thus \ [{t _ {{\ rm {back}}}} = \ frac {s} {{{c _ {{\ rm {eff}} {\ rm {, back}}}}}} \ Rightarrow {t _ {{\ rm {back}}}} = \ frac {{650 {\ rm {km}}}}} {{400 \ frac {{{\ rm {km}}}} {{\ rm {h}}}}} = 1.63 {\ rm {h}} = 1 {\ rm {h}} 38 {\ rm {min}} \] Finally, the total time is \ [t = {t _ {{\ rm {to}}}} + {t _ {{\ rm {back}}}} = 1 {\ rm {h}} 05 {\ rm {min}} + 1 {\ rm {h}} 38 {\ rm {min}} = 2 {\ rm {h}} 43 {\ rm {min}} \]